I would like to have a feedback and a different approach to my problem: I consider the embedding $W_0^{1,N}$ into the Orlicz space defined by $e^{\alpha |u|^{N/(N-1)}}$. Here $N$ is the dimension, we are on a bounded open set (say smooth) and $\alpha$ is the optimal constant such that the embedding is bounded independently of $u$ (as determined by J. Moser). This embedding is not compact. However, if we were able to "improve" the constant $\alpha$ by some $\epsilon > 0$ (meaning $e^{(\alpha + \epsilon)|u|^{N/(N-1)}}$ is bounded) then we could conclude that if $u_n \rightharpoonup u$ in $W_0^{1,N}$ then
$$e^{\alpha |u_n|^{N/(N-1)}} \rightarrow e^{\alpha |u|^{N/(N-1)}} \text{ in } L^1$$
My reasoning was to argue via Vitali convergence theorem: Since $\Omega$ is bounded, we may suppose by Rellich-Kondrachov that $u_n \rightarrow u$ almost everywhere, the other conditions are also fine. For the uniform integrability I argue via Dunford-Pettis: since by assumption we have better integrability by $\epsilon$, we may assume that $e^{(\alpha)|u_n|^{N/(N-1)}} \in L^{1+\epsilon / \alpha}$. Since we are on a finite measure space, this guarantees weak convergence in respect to $L^1$, which by Dunford-Pettis implies uniform integrability. For me, this should do the job.
However, when talking to my professor (I did not quite understand his objections), he said that there are much simpler ways, analogous to the standard Sobolev case. I would appreciate any comments on my reasoning (already I am not sure if it works) and any ideas or techniques how to reason differently.